Topological Domain Theory

Topological domain theory is
a generalization of domain theory
that includes a wider collection of topological spaces than
traditional domain theory. The generalization
overcomes certain known limitations of domain theory,
which is unable to model various awkward
combinations of computational features.

The development of topological domain theory was supported by
an EPSRC research grant "Topological Models for Computational
Metalanguages" 2003-6, which employed Matthias Schröder
and Ingo Battenfeld. Here is the final report for the
project (postscript,
pdf).

Overviews of topological domain theory

A manifesto for topological domain theory was presented in
talks at Dagstuhl and Kyoto, 2002.

Mathematical development of topological domain theory

Topological domain theory
is based on the remarkable closure properties of qcb spaces
(topological quotients of
countably based spaces).
These study of these spaces was initiated independently
by Schröder and by Menni and Simpson in the late 1990's.
Qcb spaces provide a link between general topology in mathematics,
realizability semantics in computer science, and
computability theory (especially, Weihrauch's
type two theory of effectivity).
Here is a selection of background material developing
these aspects of qcb spaces.

Topological domain theory involves restricting qcb spaces
to ones enjoying a form of chain completeness sufficient
for allowing domain-theoretic constructions.
The technical development of this is carried out in the
references below.

Computational effects in topological domain theory

The various non-functional aspects of computation
(e.g. nondeterminism, imperative features, control facilities)
are collectively known as computational effects.
Plotkin and Power have argued that many
computational effects can be modelled using free algebras
for equational theories expressing natural equalities
between efferct-triggering operations. The papers below
adapt this to topological domain theory.